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On Boolean algebras and integrally closed commutative regular rings

Published online by Cambridge University Press:  12 March 2014

Misao Nagayama
Misao Nagayama*
Affiliation:
Department of Mathematics, Tokyo Woman's Christian University, 2-6-1 Zempukuji, Suginami-Ku, Tokyo 167, Japan, E-mail: misao@tansei.cc.u-tokyo.ac.jp
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Abstract

In this paper we consider properties, related to model-completeness, of the theory of integrally closed commutative regular rings. We obtain the main theorem claiming that in a Boolean algebra B, the truth of a prenex Σn-formula whose parameters ai, partition B, can be determined by finitely many conditions built from the first entry of Tarski invariant T(ai)'s, n-characteristic D(n, ai)'s and the quantities S(ai, l) and S′(ai, l) for l < n. Then we derive two important theorems. One claims that for any Boolean algebras A and B, an embedding of A into B preserving D(n, a) for all a ϵ A is a Σn-extension. The other claims that the theory of n-separable Boolean algebras admits elimination of quantifiers in a simple definitional extension of the language of Boolean algebras. Finally we translate these results into the language of commutative regular rings.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 57 , Issue 4 , December 1992 , pp. 1305 - 1318
Copyright
Copyright © Association for Symbolic Logic 1992

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References

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